Fault-tolerant hamiltonian laceability of hypercubes

It is known that every hypercube Qn is a bipartite graph. Assume that n 2 and F is a subset of edges with |F | n− 2. We prove that there exists a hamiltonian path in Qn − F between any two vertices of different partite sets. Moreover, there exists a path of length 2n − 2 between any two vertices of the same partite set. Assume that n 3 and F is a subset of edges with |F | n− 3. We prove that there exists a hamiltonian path in Qn − {v} −F between any two vertices in the partite set without v. Furthermore, all bounds are tight.  2002 Elsevier Science B.V. All rights reserved.